Coordinating concurrent transmissions: A constant-factor - PowerPoint PPT Presentation

Coordinating concurrent transmissions: A constant-factor approximation of maximum-weight independent set in local conflict graphs Petteri Kaski 1 Aleksi Penttinen 2 Jukka Suomela 1 1 HIIT University of Helsinki Finland 2 Networking Laboratory Helsinki Univ. of Technology Finland AdHoc-NOW 24 September 2007

Wireless communication links 2 / 14

Interference — near-far effect 3 / 14

Conflict graph 4 / 14

Independent set in conflict graph 5 / 14

Independent set in conflict graph 6 / 14

Problems associated with conflict graph Problems: 1 2 time units A. Given per-link weights, 1 2 time units find maximum-weight independent set = nonconflicting set of links 1 2 time units B. Given per-link data transmission needs, 1 2 time units find a minimum-length link schedule (generalisation of 1 2 time units fractional graph colouring) 7 / 14

Known properties Problems: 1 2 time units A. max-weight independent set 1 2 time units B. min-length link schedule Known properties: 1 2 time units ◮ Approximation of A 1 2 time units implies approximation of B (e.g., Young 2001, Jansen 2003) ◮ A and B hard to solve and 1 2 time units approximate in general graphs (Lund–Yannakakis 1994, H˚ astad 1999, Khot 2001) 8 / 14

Assumptions on the problem structure Define a new family of graphs: N -local conflict graphs ok ok Assumptions: ◮ bounded density of devices : not ok at most N devices in unit disk ok ◮ bounded range of interference : interfering transmitter must be close to interfered receiver Non-assumptions: not ok ◮ interference occurs if. . . > N 9 / 14

N -local conflict graphs: large family Contains, for example: ◮ bipartite graphs ok ◮ cycles ok ◮ complete graph on N 2 vertices not ok ◮ subgraphs of ok N -local conflict graphs ◮ ( N − 1)-local conflict graphs Generalisation of local graphs and civilised graphs not ok > N 10 / 14

N -local conflict graphs: new family Not contained in: ◮ bounded-degree graphs ok (proof: K 1 , n ) ok ◮ bounded-density graphs (proof: K n , n ) not ok ◮ planar graphs (proof: K 3 , 3 ) ok ◮ disk graphs (proof: K 3 , 3 ) ◮ bipartite graphs (proof: K 3 ) ◮ graphs closed under taking minors not ok > N (proof: split edges of a large clique) 11 / 14

Our results Problems: A. max-weight independent set ok B. min-length link schedule ok Assumptions: not ok ◮ bounded density of devices ok ◮ bounded range of interference Our results: ◮ A & B : (5 + ǫ )-approximation not ok ◮ A & B : no PTAS > N 12 / 14

Algorithm sketch Apply a shifting strategy 0 1 2 0 1 2 0 1 2 0 0 (Hochbaum–Maass 1985) 1 2 0 ◮ Use a modular grid 1 2 0 ◮ Try several locations ◮ Choose the best Short links (within a grid cell): ◮ Exhaustive search for each cell senders receivers Long links (between grid cells): ◮ Find a large directed cut 13 / 14

Summary ◮ N -local conflict graphs: 0 1 2 0 1 2 0 1 2 0 0 1 ◮ bounded density of devices, 2 0 bounded range of interference 1 2 0 ◮ Max-weight independent set, min-length link schedule ◮ (5 + ǫ )-approximation, no PTAS ◮ Open problems: distributed, coordinate-free algorithms? senders receivers http://www.hiit.fi/ada/geru jukka.suomela@cs.helsinki.fi 14 / 14